The septenary numeral system

Numeral system

A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....

 is the base- number system, and uses the digits 0-6.

Multiplication table

1	2	3	4	5	6	10
2	4	6	11	13	15	20
3	6	12	15	21	24	30
4	11	15	22	26	33	40
5	13	21	26	34	42	50
6	15	24	33	42	51	60
10	20	30	40	50	60	100

Fractions

Fractions expressed in septenary will repeat a sequence of digits unless the denominator is a power of seven. Few fractions can be expressed in a finite number of digits:

Decimal	Septimal (periodic part)
1/2	1/2 = 0.3...
1/3	1/3 = 0.2...
1/4	1/4 = 0.15...
1/5	1/5 = 0.1254...
1/6	1/6 = 0.1...
1/7	1/10 = 0.1
1/8	1/11 = 0.06...
1/9	1/12 = 0.053...
1/10	1/13 = 0.0462...
1/12	1/15 = 0.04...
1/14	1/20 = 0.03...
1/15	1/21 = 0.0316...
1/16	1/22 = 0.03...
1/18	1/24 = 0.025...
1/19	1/25 = 0.024...
1/20	1/26 = 0.0231...
1/21	1/30 = 0.02...
1/24	1/33 = 0.02...
...	...
1/49	1/100 = 0.01

Irrational Numbers

Algebraic irrational number		In decimal	In septenary
√2 Square root of 2 The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...	(the length of the diagonal Diagonal A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...  of a unit square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... )	1.41421356237309... (≈ 1.414)	1.262034545211232611 ... (≈ 1.262)
√3 Square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property...	(the length of the diagonal of a unit cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... , or twice the height Height Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example "The height of the building is 50 m" or "The height of the airplane is 10,000 m"...  of an equilateral triangle of unit side)	1.73205080756887... (≈ 1.732)	1.506044021410166456 ... (≈ 1.506 ≈ 1.5)
√5	(the length of the diagonal Diagonal A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...  of a 1×2 rectangle Rectangle In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle... )	2.2360679774997... (≈ 2.236)	2.143654106250351 ... (≈ 2.144)
φ Golden ratio In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...	(phi, the golden ratio = )	1.6180339887498... (≈ 1.618)	1.42166203646016... (≈ 1.422)
Transcendental irrational number		In decimal	In septenary
π Pi ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...	(pi, the ratio of circumference Circumference The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....  to diameter Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle... )	3.1415926535898... (≈ 3.1416)	3.06636514320361341... (≈ 3.1)
e E (mathematical constant) The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...	(the base of the natural logarithm Natural logarithm The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828... )	2.718281828459045... (≈ 2.718)	2.50124106542265... (≈ 2.5)

One feature of this system is that 3.1 (= 22/7) approximates

Pi

' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

 with a relative error of 0.04 %.

In fiction

• The Tau
  Tau (Warhammer 40,000)
  In the universe of Games Workshop's table-top wargame Warhammer 40,000, the Tau Empire is an alien race, inhabiting a small but dense region of space on the eastern edge of the Galaxy, roughly 300 light years in diameter...
  
   of Sci-fi Table-top battle game Warhammer 40,000
  Warhammer 40,000
  Warhammer 40,000 is a tabletop miniature wargame produced by Games Workshop, set in a dystopian science fantasy universe. Warhammer 40,000 was created by Rick Priestley in 1987 as the futuristic companion to Warhammer Fantasy Battle, sharing many game mechanics...
  
  use a base-7 counting system.

• In the New Series Adventures
  New Series Adventures (Doctor Who)
  The New Series Adventures are a series of novels relating to the long-running BBC science fiction television series, Doctor Who. The 'NSAs', as they are often referred to, are published by BBC Books, and are regularly published in the spring and autumn of each year. Beginning and concluding with...
  
  of Doctor Who
  Doctor Who
  Doctor Who is a British science fiction television programme produced by the BBC. The programme depicts the adventures of a time-travelling humanoid alien known as the Doctor who explores the universe in a sentient time machine called the TARDIS that flies through time and space, whose exterior...
  
  , the time lord
  Time Lord
  The Time Lords are an ancient extraterrestrial race and civilization of humanoids in the British science fiction television series Doctor Who, of which the series' eponymous protagonist, the Doctor, is a member...
  
  s employ a number system using base 7.

• The Halo 3 Alternate Reality Game "IRIS" used a countdown clock in base-7.

• In the Halo
  Halo (series)
  Halo is a multi-million dollar science fiction video game franchise created by Bungie and now managed by 343 Industries and owned by Microsoft Studios. The series centers on an interstellar war between humanity and a theocratic alliance of aliens known as the Covenant...
  
   videogame series, the Forerunners use a base-7 number system.

• In the online RPG Kingdom of Loathing
  Kingdom of Loathing
  Kingdom of Loathing is a browser-based, multiplayer role-playing game designed and operated by Asymmetric Publications, including creator Zack "Jick" Johnson and writer Josh "Mr. Skullhead" Nite. The game was released in 2003...
  
  , the Dwarven miners use a base-7 number system.